A sample complexity separation between non-convex and convex meta-learning

One popular trend in meta-learning is to learn from many training tasks a common initialization that a gradient-based method can use to solve a new task with few samples. The theory of metalearning is still in its early stages, with several recent learning-theoretic analyses of methods such as Reptile (Nichol et al., 2018) being for convex models. This work shows that convex-case analysis might be insufficient to understand the success of meta-learning, and that even for non-convex models it is important to look inside the optimization black-box, specifically at properties of the optimization trajectory. We construct a simple meta-learning instance that captures the problem of one-dimensional subspace learning. For the convex formulation of linear regression on this instance, we show that the new task sample complexity of any initialization-based meta-learning algorithm is Ω(d), where d is the input dimension. In contrast, for the non-convex formulation of a two layer linear network on the same instance, we show that both Reptile and multi-task representation learning can have new task sample complexity of O(1), demonstrating a separation from convex meta-learning. Crucially, analyses of the training dynamics of these methods reveal that they can meta-learn the correct subspace onto which the data should be projected.